Find the Roots/Zeros Using the Rational Roots Test x^3+5x^2-73x-77

$x^{3} + 5 x^{2} - 73 x - 77$

Step 1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 7, \pm 11, \pm 77$

$q = \pm 1$

Step 2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 7, \pm 11, \pm 77$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(- 1)}^{3} + 5 {(- 1)}^{2} - 73 \cdot - 1 - 77$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - 1$ is a root of the polynomial.

Simplify each term.

Raise $- 1$ to the power of $3$ .

$- 1 + 5 {(- 1)}^{2} - 73 \cdot - 1 - 77$

Raise $- 1$ to the power of $2$ .

$- 1 + 5 \cdot 1 - 73 \cdot - 1 - 77$

Multiply $5$ by $1$ .

$- 1 + 5 - 73 \cdot - 1 - 77$

Multiply $- 73$ by $- 1$ .

$- 1 + 5 + 73 - 77$

Simplify by adding and subtracting.

Add $- 1$ and $5$ .

$4 + 73 - 77$

Add $4$ and $73$ .

$77 - 77$

Subtract $77$ from $77$ .

$0$

Step 5

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 5 x^{2} - 73 x - 77}{x + 1}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$- 1$	$1$	$5$	$- 73$	$- 77$
	\xa0			\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$- 1$	$1$	$5$	$- 73$	$- 77$
	\xa0			\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(- 1)$ and place the result of $(- 1)$ under the next term in the dividend $(5)$ .

$- 1$	$1$	$5$	$- 73$	$- 77$
	\xa0	$- 1$		\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$5$	$- 73$	$- 77$
	\xa0	$- 1$		\xa0
	$1$	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(- 1)$ and place the result of $(- 4)$ under the next term in the dividend $(- 73)$ .

$- 1$	$1$	$5$	$- 73$	$- 77$
	\xa0	$- 1$	$- 4$	\xa0
	$1$	$4$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$5$	$- 73$	$- 77$
	\xa0	$- 1$	$- 4$	\xa0
	$1$	$4$	$- 77$

Multiply the newest entry in the result $(- 77)$ by the divisor $(- 1)$ and place the result of $(77)$ under the next term in the dividend $(- 77)$ .

$- 1$	$1$	$5$	$- 73$	$- 77$
	\xa0	$- 1$	$- 4$	$77$
	$1$	$4$	$- 77$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$- 1$	$1$	$5$	$- 73$	$- 77$
	\xa0	$- 1$	$- 4$	$77$
	$1$	$4$	$- 77$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{2} + 4 x - 77$

Simplify the quotient polynomial.

$x^{2} + 4 x - 77$

Step 7

Factor $x^{2} + 4 x - 77$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 77$ and whose sum is $4$ .

$- 7, 11$

Write the factored form using these integers.

$(x + 1) + (x - 7) (x + 11)$

$(x + 1) (x - 7) (x + 11)$

Step 8

Factor the left side of the equation.

Factor $x^{3} + 5 x^{2} - 73 x - 77$ using the rational roots test.

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 77, \pm 7, \pm 11 q = \pm 1$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 77, \pm 7, \pm 11$

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} + 5 {(- 1)}^{2} - 73 \cdot - 1 - 77$

Raise $- 1$ to the power of $3$ .

$- 1 + 5 {(- 1)}^{2} - 73 \cdot - 1 - 77$

Raise $- 1$ to the power of $2$ .

$- 1 + 5 \cdot 1 - 73 \cdot - 1 - 77$

Multiply $5$ by $1$ .

$- 1 + 5 - 73 \cdot - 1 - 77$

Add $- 1$ and $5$ .

$4 - 73 \cdot - 1 - 77$

Multiply $- 73$ by $- 1$ .

$4 + 73 - 77$

Add $4$ and $73$ .

$77 - 77$

Subtract $77$ from $77$ .

$0$

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 5 x^{2} - 73 x - 77}{x + 1}$

Divide $x^{3} + 5 x^{2} - 73 x - 77$ by $x + 1$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{3}$

$5 x^{2}$

$73 x$

$77$

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$
			+	$x^{3}$	+	$x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + x^{2}$

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$
			-	$x^{3}$	-	$x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$73 x$

Divide the highest order term in the dividend $4 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$4 x$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$73 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$4 x$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$73 x$
					+	$4 x^{2}$	+	$4 x$

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{2} + 4 x$

				$x^{2}$	+	$4 x$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$73 x$
					-	$4 x^{2}$	-	$4 x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$4 x$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$73 x$
					-	$4 x^{2}$	-	$4 x$
							-	$77 x$

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	+	$4 x$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$73 x$
					-	$4 x^{2}$	-	$4 x$
							-	$77 x$	-	$77$

Divide the highest order term in the dividend $- 77 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$4 x$	-	$77$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$73 x$
					-	$4 x^{2}$	-	$4 x$
							-	$77 x$	-	$77$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$4 x$	-	$77$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$73 x$
					-	$4 x^{2}$	-	$4 x$
							-	$77 x$	-	$77$
							-	$77 x$	-	$77$

The expression needs to be subtracted from the dividend, so change all the signs in $- 77 x - 77$

				$x^{2}$	+	$4 x$	-	$77$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$73 x$
					-	$4 x^{2}$	-	$4 x$
							-	$77 x$	-	$77$
							+	$77 x$	+	$77$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$4 x$	-	$77$
$x$	+	$1$		$x^{3}$	+	$5 x^{2}$	-	$73 x$	-	$77$
			-	$x^{3}$	-	$x^{2}$
					+	$4 x^{2}$	-	$73 x$
					-	$4 x^{2}$	-	$4 x$
							-	$77 x$	-	$77$
							+	$77 x$	+	$77$
										$0$

Since the remander is $0$ , the final answer is the quotient.

$x^{2} + 4 x - 77$

Write $x^{3} + 5 x^{2} - 73 x - 77$ as a set of factors.

$(x + 1) (x^{2} + 4 x - 77) = 0$

Factor $x^{2} + 4 x - 77$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 77$ and whose sum is $4$ .

$- 7, 11$

Write the factored form using these integers.

$(x + 1) ((x - 7) (x + 11)) = 0$

Remove unnecessary parentheses.

$(x + 1) (x - 7) (x + 11) = 0$

Step 9

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$x - 7 = 0$

$x + 11 = 0$

Step 10

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 11

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Step 12

Set $x + 11$ equal to $0$ and solve for $x$ .

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

Subtract $11$ from both sides of the equation.

$x = - 11$

Step 13

The final solution is all the values that make $(x + 1) (x - 7) (x + 11) = 0$ true.

$x = - 1, 7, - 11$

Step 14

Find the Roots/Zeros Using the Rational Roots Test x^3+5x^2-73x-77

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